Sparse image representations with shift and rotation invariance constraints

This paper presents a sparse image representation and its dictionary learning under shift and rotation invariance constraints. Sparse coding is a generative signal model that approximates signals by linear combinations of atoms in a dictionary. Since a sparsity penalty is introduced during signal approximation and dictionary learning, the dictionary represents primal structures of the signals. Under the shift and rotation invariance, the atoms in the dictionary are generated by the rotation and translation of the two-dimensional basic functions that indicate primal local structures of an image. The number of atoms for representation of an image is a product of the numbers of translation positions, rotation angles and basic functions. For the decomposition and dictionary learning, the huge storage capacity is required to store the coefficients, which are assigned to the atoms. In order to reduce the number of the coefficients and the computational burden, we propose a restricted image generative model for the shift and rotation invariant sparse representation. In experiment, the dictionary learning for synthetic and natural images is demonstrated. The results show that the sparse decomposition using the dictionary learnt by the proposed method can decompose images into parts which have different features.